Torus T^3 - Knowlpedia

the real Lie group $\mathbb T^{3}$ is $U(1)^{3}$ with coordinatewise multiplication; equivalently it is $\mathbb R^{3}/(2\pi\mathbb Z)^{3}$ with addition.

Dimensions and structure

This group has real dimension $3$ . It is compact, connected and abelian. Its universal covering group is additive $\mathbb R^{3}$ . The complex multiplicative group has a different real dimension and is noncompact.

Direct verification

Coordinatewise multiplication and conjugation preserve the unit-circle equations. Each circle contributes one real dimension; the exponential map from Rⁿ has kernel 2πZⁿ.